Armature doesn't move part of the Mesh
It is difficult to say for sure without the file, but it looks like the join did not work correctly for some reason. Select the object and go into edit mode. Do the pieces that are not moving go into edit mode as well (you can see the vertices) or do they remain in object mode? If they remain in object mode, they are not connected to the mesh. Try re-joining them.

The other most likely possibility is that those pieces are weighted with a weight of 0 with regards to the armature. This could have happened if the pieces were joined after you parented the armature. If it was an automatic parent, try parenting again. If it was done manually, set the weights for the new vertices.

I hope one of these solves your problem.